Date: Dec 8, 2012 9:06 PM
Author: fom
Subject: Re: fom - 01 - preface
On 12/8/2012 1:49 PM, WM wrote:> On 8 Dez., 19:16, fom <fomJ...@nyms.net> wrote:>> On 12/8/2012 9:08 AM, WM wrote:>>>> There are certain ongoing investigations>> into the structure of mathematical proofs>> that interpret the linguistic usage differently>> from "mathematical logic". You would be>> looking for various discussions of>> context-dependent quantification where it>> is being related to mathematical usage.>>>> You will find that a statment such as>>>> "Fix x">>>> followed by>>>> "Let y be chosen distinct from x">>>> is interpreted relative to two>> different domains of discourse.>>>> This is just how one would imagine>> traversing from the bottom of a>> partition lattice.>> A question: Do you believe that there are more than countably many> finite words?> Do you believe that you can use infinite words (not finite> descriptions of infinite sequences).> Do you believe that you can put in order what you cannot distinguish?>> Regards, WM>

There is a certain history here.

As set theory developed, Cantor was confrontedwith the notion of "absolute infinity".

I prefer to go with Kant:

"Infinity is plurality without unity"

and interpret the objects spoken of in typicaldiscussions of set theory as transfinite numbers.

As for "unity", Cantor wrote thefollowing in his criticism of Frege:

"...to take 'the extension of a concept' as thefoundation of the number-concept. He [Frege]overlooks the fact that in general the 'extensionof a concept' is something quantitatively completelyundetermined. Only in certain cases is the 'extensionof a concept' quantitatively determined, thenit certainly has, if it is finite, a definitenatural number, and if infinite, a definite power.For such quantitative determination of the'extension of a concept' the concepts 'number'and 'power' must previously be already givenfrom somewhere else, and it is a reversal ofthe proper order when one undertakes to basethe latter concepts on the concept 'extensionof a concept'."

Cantor's transfinite sequences begin by simplymaking precise the natural language referencesto the natural numbers as a definite whole. And,he justifies his acceptance of the transfinitewith remarks such as:

"... the potential infinite is only anauxiliary or relative (or relational)concept, and always indicates an underlyingtransfinite without which it can neitherbe nor be thought."

But the question of existence speaks preciselyto the first edition of "Principia Mathematica"by Russell & Whitehead. I would love to havethe time to revisit what has been done there.

Russell's first version had been guided inlarge part by his views on denotation. So,the presupposition failure inherent to referencewas to be addressed by his description theory.Given that, he ultimately would be committedto the axiom of reducibility.

It is interesting to read what he saysconcerning that axiom and set existence,

"The axiom of reducibility is evenmore essential in the theory ofclasses. It should be observed,in the first place, that if we assumethe existence of classes, the axiomof reducibility can be proved. For inthat case, given any function phi..z^of whatever order, there is a class Aconsisting of just those objects whichsatisfy phi..z^. Hence, "phi(x)" isequivalent to "x belongs to A". But,"x belongs to A" is a statement containingno apparent variable, and is thereforea predicative function of x. Hence, ifwe assume the existence of classes, theaxiom of reducibility becomes unnecessary."

Personally, I do not think he shouldhave given it up.

As for my personal beliefs, I reject, forthe most part, the ontological presuppositionsof modern logicians so far as I can discern themfrom what I read. Frege made a great achievementin recognizing how to formulate a deductivecalculus for mathematics. But, I side withAristotle on the nature of what roles are playedby a deductive calculus. Scientific demonstrationis distinct from dialectical argumentation thatargues from belief. In turn, that distinctioninforms that a scientific language is built upsynthetically. The objects of that languageare individually described using definitions.The objects of that language are individuallypresumed to exist. Consequently, thenames which complete the "incomplete symbols"exist as references only by virtue of the factthat the first names introduced for use in thescience are a well-ordered sequence.

Since I cannot possibly defend introducingmore than some finite number of names inthis fashion, the assumption of transfinitenumbers in set theory has a consequence. Itcan be reconciled with this position onlyif models of set theory are admissible assuch when they have a global well-ordering.

The largest transitive model of ZFC set theorywith these properties is HOD (hereditarilyordinal definable).